$\left(\dfrac{y\pm 7x}{2}\right)^2+7\left(\dfrac{y \mp x}{2}\right)^2=2(y^2+7x^2)$ So, for $n=3,y_1=x_1=1$ and If $y_k \equiv x_k \pmod 4,\;\; y_{k+1}=\dfrac{|y_k-7x_k|}{2},\;\;x_{k+1}=\dfrac{y+x}{2}$ If $y_k \not \equiv x_k \pmod 4,\;\; y_{k+1}=\dfrac{y_k+7x_k}{2},\;\;x_{k+1}=\dfrac{|y-x|}{2}$ $x_{k+1},y_{k+1}$ are odd too.